That’s not unique to cheap airbrushes, as the trigger is usually only really held in place by the needle, though it may have some additional protrusions to keep it inside the
airbrush a bit. On a Badger, for example, pulling the needle out and holding the
airbrush upside-down is enough to lose the trigger — and then you’ll have great fun getting it back in because a sprung bit of steel sits against the back of it, that you somehow need to push back before you can re-insert it. Iwatas are similar but at least have a kind of pivots on the trigger so it needs to be rotated 90° before it can come out of its slot. Doesn’t stop it coming up too far, of course, but that’s not as bad as it dropping out unexpectedly.
As for masking tape: buy the kind aimed at modellers, or at least the yellow or orange or otherwise colourful kind sold at DIY stores. The old-fashioned white type, like Jim said, is far too adhesive.